r"""
.. _qubit_rotation:

Basic tutorial: qubit rotation
==============================

.. meta::
    :property="og:description": To see how PennyLane allows the easy construction and optimization
        of quantum functions, let's consider the 'hello world' of QML: qubit rotation.
    :property="og:image": https://pennylane.ai/qml/_images/bloch.png

.. related::

   plugins_hybrid Plugins and hybrid computation
   tutorial_gaussian_transformation Gaussian transformation
   tutorial_state_preparation Training a quantum circuit with PyTorch

*Author: Josh Izaac — Posted: 11 October 2019. Last updated: 19 January 2021.*

To see how PennyLane allows the easy construction and optimization of quantum functions, let's
consider the simple case of **qubit rotation** the PennyLane version of the 'Hello, world!'
example.

The task at hand is to optimize two rotation gates in order to flip a single
qubit from state :math:`\left|0\right\rangle` to state :math:`\left|1\right\rangle`.


The quantum circuit
-------------------

In the qubit rotation example, we wish to implement the following quantum circuit:

.. figure:: ../demonstrations/qubit_rotation/rotation_circuit.png
    :align: center
    :width: 40%
    :target: javascript:void(0);

Breaking this down step-by-step, we first start with a qubit in the ground state
:math:`|0\rangle = \begin{bmatrix}1 & 0 \end{bmatrix}^T`,
and rotate it around the x-axis by applying the gate

.. math::
    R_x(\phi_1) = e^{-i \phi_1 \sigma_x /2} =
    \begin{bmatrix} \cos \frac{\phi_1}{2} &  -i \sin \frac{\phi_1}{2} \\
                   -i \sin \frac{\phi_1}{2} &  \cos \frac{\phi_1}{2}
    \end{bmatrix},

and then around the y-axis via the gate

.. math::
    R_y(\phi_2) = e^{-i \phi_2 \sigma_y/2} =
   \begin{bmatrix} \cos \frac{\phi_2}{2} &  - \sin \frac{\phi_2}{2} \\
                   \sin \frac{\phi_2}{2} &  \cos \frac{\phi_2}{2}
   \end{bmatrix}.

After these operations the qubit is now in the state

.. math::  | \psi \rangle = R_y(\phi_2) R_x(\phi_1) | 0 \rangle.

Finally, we measure the expectation value :math:`\langle \psi \mid \sigma_z \mid \psi \rangle`
of the Pauli-Z operator

.. math::
   \sigma_z =
   \begin{bmatrix} 1 &  0 \\
                   0 & -1
   \end{bmatrix}.

Using the above to calculate the exact expectation value, we find that

.. math::
    \langle \psi \mid \sigma_z \mid \psi \rangle
    = \langle 0 \mid R_x(\phi_1)^\dagger R_y(\phi_2)^\dagger \sigma_z  R_y(\phi_2) R_x(\phi_1) \mid 0 \rangle
    = \cos(\phi_1)\cos(\phi_2).

Depending on the circuit parameters :math:`\phi_1` and :math:`\phi_2`, the
output expectation lies between :math:`1` (if :math:`\left|\psi\right\rangle = \left|0\right\rangle`)
and :math:`-1` (if :math:`\left|\psi\right\rangle = \left|1\right\rangle`).
"""

##############################################################################
#
# Let's see how we can easily implement and optimize this circuit using PennyLane.
#
# Importing PennyLane and NumPy
# -----------------------------
#
# The first thing we need to do is import PennyLane, as well as the wrapped version
# of NumPy provided by PennyLane.

import pennylane as qml
from pennylane import numpy as np


##############################################################################
# .. important::
#
#     When constructing a hybrid quantum/classical computational model with PennyLane,
#     it is important to **always import NumPy from PennyLane**, not the standard NumPy!
#
#     By importing the wrapped version of NumPy provided by PennyLane, you can combine
#     the power of NumPy with PennyLane:
#
#     * continue to use the classical NumPy functions and arrays you know and love
#     * combine quantum functions (evaluated on quantum hardware/simulators) and
#       classical functions (provided by NumPy)
#     * allow PennyLane to automatically calculate gradients of both classical and
#       quantum functions


##############################################################################
# Creating a device
# -----------------
#
# Before we can construct our quantum node, we need to initialize a **device**.
#
# .. admonition:: Definition
#     :class: defn
#
#     Any computational object that can apply quantum operations and return a measurement value
#     is called a quantum **device**.
#
#     In PennyLane, a device could be a hardware device (such as the IBM QX4, via the
#     PennyLane-PQ plugin), or a software simulator (such as Strawberry Fields, via the
#     PennyLane-SF plugin).
#
# .. tip::
#
#    *Devices are loaded in PennyLane via the function* :func:`~.pennylane.device`
#
#
# PennyLane supports devices using both the qubit model of quantum computation and devices
# using the CV model of quantum computation. In fact, even a hybrid computation containing
# both qubit and CV quantum nodes is possible; see the
# :ref:`hybrid computation example <hybrid_computation_example>` for more details.
#
# For this tutorial, we are using the qubit model, so let's initialize the ``'default.qubit'`` device
# provided by PennyLane; a simple pure-state qubit simulator.

dev1 = qml.device("default.qubit", wires=1)

##############################################################################
# For all devices, :func:`~.pennylane.device` accepts the following arguments:
#
# * ``name``: the name of the device to be loaded
# * ``wires``: the number of subsystems to initialize the device with
#
# Here, as we only require a single qubit for this example, we set ``wires=1``.

##############################################################################
# Constructing the QNode
# ----------------------
#
# Now that we have initialized our device, we can begin to construct a
# **quantum node** (or QNode).
#
#
# .. admonition:: Definition
#     :class: defn
#
#     QNodes are an abstract encapsulation of a quantum function, described by a
#     quantum circuit. QNodes are bound to a particular quantum device, which is
#     used to evaluate expectation and variance values of this circuit.
#
# .. tip::
#
#    *QNodes can be constructed via the* :class:`~.pennylane.QNode`
#    *class, or by using the provided* :func:`~.pennylane.qnode` decorator.
#
# First, we need to define the quantum function that will be evaluated in the QNode:


def circuit(params):
    qml.RX(params[0], wires=0)
    qml.RY(params[1], wires=0)
    return qml.expval(qml.PauliZ(0))


##############################################################################
# This is a simple circuit, matching the one described above.
# Notice that the function ``circuit()`` is constructed as if it were any
# other Python function; it accepts a positional argument ``params``, which may
# be a list, tuple, or array, and uses the individual elements for gate parameters.
#
# However, quantum functions are a **restricted subset** of Python functions.
# For a Python function to also be a valid quantum function, there are some
# important restrictions:
#
# * **Quantum functions must contain quantum operations, one operation per line,
#   in the order in which they are to be applied.**
#
#   In addition, we must always specify the subsystem the operation applies to,
#   by passing the ``wires`` argument; this may be a list or an integer, depending
#   on how many wires the operation acts on.
#
#   For a full list of quantum operations, see :doc:`the documentation <introduction/operations>`.
#
# * **Quantum functions must return either a single or a tuple of measured observables**.
#
#   As a result, the quantum function always returns a classical quantity, allowing
#   the QNode to interface with other classical functions (and also other QNodes).
#
#   For a full list of observables, see :doc:`the documentation <introduction/operations>`.
#   The documentation also provides details on supported :doc:`measurement return types <introduction/measurements>`.
#
# .. note::
#
#     Certain devices may only support a subset of the available PennyLane
#     operations/observables, or may even provide additional operations/observables.
#     Please consult the documentation for the plugin/device for more details.
#
# Once we have written the quantum function, we convert it into a :class:`~.pennylane.QNode` running
# on device ``dev1`` by applying the :func:`~.pennylane.qnode` decorator.
# **directly above** the function definition:


@qml.qnode(dev1, interface="autograd")
def circuit(params):
    qml.RX(params[0], wires=0)
    qml.RY(params[1], wires=0)
    return qml.expval(qml.PauliZ(0))


##############################################################################
# Thus, our ``circuit()`` quantum function is now a :class:`~.pennylane.QNode`, which will run on
# device ``dev1`` every time it is evaluated.
#
# To evaluate, we simply call the function with some appropriate numerical inputs:

print(circuit([0.54, 0.12]))

##############################################################################
# Calculating quantum gradients
# -----------------------------
#
# The gradient of the function ``circuit``, encapsulated within the ``QNode``,
# can be evaluated by utilizing the same quantum
# device (``dev1``) that we used to evaluate the function itself.
#
# PennyLane incorporates both analytic differentiation, as well as numerical
# methods (such as the method of finite differences). Both of these are done
# automatically.
#
# We can differentiate by using the built-in :func:`~.pennylane.grad` function.
# This returns another function, representing the gradient (i.e., the vector of
# partial derivatives) of ``circuit``. The gradient can be evaluated in the same
# way as the original function:

dcircuit = qml.grad(circuit, argnum=0)

##############################################################################
# The function :func:`~.pennylane.grad` itself **returns a function**, representing
# the derivative of the QNode with respect to the argument specified in ``argnum``.
# In this case, the function ``circuit`` takes one argument (``params``), so we
# specify ``argnum=0``. Because the argument has two elements, the returned gradient
# is two-dimensional. We can then evaluate this gradient function at any point in the parameter space.

print(dcircuit([0.54, 0.12]))

################################################################################
# **A note on arguments**
#
# Quantum circuit functions, being a restricted subset of Python functions,
# can also make use of multiple positional arguments and keyword arguments.
# For example, we could have defined the above quantum circuit function using
# two positional arguments, instead of one array argument:


@qml.qnode(dev1, interface="autograd")
def circuit2(phi1, phi2):
    qml.RX(phi1, wires=0)
    qml.RY(phi2, wires=0)
    return qml.expval(qml.PauliZ(0))


################################################################################
# When we calculate the gradient for such a function, the usage of ``argnum``
# will be slightly different. In this case, ``argnum=0`` will return the gradient
# with respect to only the first parameter (``phi1``), and ``argnum=1`` will give
# the gradient for ``phi2``. To get the gradient with respect to both parameters,
# we can use ``argnum=[0,1]``:

dcircuit = qml.grad(circuit2, argnum=[0, 1])
print(dcircuit(0.54, 0.12))

################################################################################
# Keyword arguments may also be used in your custom quantum function. PennyLane
# does **not** differentiate QNodes with respect to keyword arguments,
# so they are useful for passing external data to your QNode.


################################################################################
# Optimization
# ------------
#
# .. admonition:: Definition
#     :class: defn
#
#     If using the default NumPy/Autograd interface, PennyLane provides a collection
#     of optimizers based on gradient descent. These optimizers accept a cost function
#     and initial parameters, and utilize PennyLane's automatic differentiation
#     to perform gradient descent.
#
# .. tip::
#
#    *See* :doc:`introduction/optimizers` *for details and documentation of available optimizers*
#
# Next, let's make use of PennyLane's built-in optimizers to optimize the two circuit
# parameters :math:`\phi_1` and :math:`\phi_2` such that the qubit, originally in state
# :math:`\left|0\right\rangle`, is rotated to be in state :math:`\left|1\right\rangle`. This is equivalent to measuring a
# Pauli-Z expectation value of :math:`-1`, since the state :math:`\left|1\right\rangle` is an eigenvector
# of the Pauli-Z matrix with eigenvalue :math:`\lambda=-1`.
#
# In other words, the optimization procedure will find the weights
# :math:`\phi_1` and :math:`\phi_2` that result in the following rotation on the Bloch sphere:
#
# .. figure:: ../demonstrations/qubit_rotation/bloch.png
#     :align: center
#     :width: 70%
#     :target: javascript:void(0);
#
# To do so, we need to define a **cost** function. By *minimizing* the cost function, the
# optimizer will determine the values of the circuit parameters that produce the desired outcome.
#
# In this case, our desired outcome is a Pauli-Z expectation value of :math:`-1`. Since we
# know that the Pauli-Z expectation is bound between :math:`[-1, 1]`, we can define our
# cost directly as the output of the QNode:


def cost(x):
    return circuit(x)


################################################################################
# To begin our optimization, let's choose small initial values of :math:`\phi_1` and :math:`\phi_2`:

init_params = np.array([0.011, 0.012], requires_grad=True)
print(cost(init_params))

################################################################################
# We can see that, for these initial parameter values, the cost function is close to :math:`1`.
#
# Finally, we use an optimizer to update the circuit parameters for 100 steps. We can use the built-in
# :class:`~.pennylane.GradientDescentOptimizer` class:

# initialise the optimizer
opt = qml.GradientDescentOptimizer(stepsize=0.4)

# set the number of steps
steps = 100
# set the initial parameter values
params = init_params

for i in range(steps):
    # update the circuit parameters
    params = opt.step(cost, params)

    if (i + 1) % 5 == 0:
        print("Cost after step {:5d}: {: .7f}".format(i + 1, cost(params)))

print("Optimized rotation angles: {}".format(params))

################################################################################
# We can see that the optimization converges after approximately 40 steps.
#
# Substituting this into the theoretical result :math:`\langle \psi \mid \sigma_z \mid \psi \rangle = \cos\phi_1\cos\phi_2`,
# we can verify that this is indeed one possible value of the circuit parameters that
# produces :math:`\langle \psi \mid \sigma_z \mid \psi \rangle=-1`, resulting in the qubit being rotated
# to the state :math:`\left|1\right\rangle`.
#
# .. note::
#
#     Some optimizers, such as :class:`~.pennylane.AdagradOptimizer`, have
#     internal hyperparameters that are stored in the optimizer instance. These can
#     be reset using the :meth:`reset` method.
#
# Continue on to the next tutorial, :ref:`gaussian_transformation`, to see a similar example using
# continuous-variable (CV) quantum nodes.
#
#
# About the author
# ----------------
# .. include:: ../_static/authors/josh_izaac.txt